EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER Abstract. We prove, for certain pairs G, G0of finite groups of Lie type,* * that the p-fusion systems Fp(G) and Fp(G0) are equivalent. In other words, th* *ere is an isomorphism between a Sylow p-subgroup of G and one of G0 which preserves p-fusion. This occurs, for example, when G = G(q) and G0= G(q0) for a simple Lie "type" G, and q and q0 are prime powers, both prime to p, which generate the same closed subgroup of p-adic units. Our proof us* *es homotopy theoretic properties of the p-completed classifying spaces of G* * and G0, and we know of no purely algebraic proof of this result. When G is a finite group and p is a prime, the fusion system Fp(G) is the category whose objects are the p-subgroups of G, and whose morphisms are the homomorphisms between subgroups induced by conjugation in G. If G0is another finite group, then Fp(G) and Fp(G0) are isotypically equivalent if there is an * *equiv- alence of categories between them which commutes, up to natural isomorphism of functors, with the forgetful functors from Fp(-) to the category of groups. Alt* *er- natively, Fp(G) and Fp(G0) are isotypically equivalent if there is an isomorphi* *sm between Sylow p-subgroups of G and of G0which is "fusion preserving" in the sen* *se of Definition 1.2 below. The goal of this paper is to use methods from homotopy theory to prove that certain pairs of fusion systems of finite groups of Lie type are isotypically e* *quivalent. Our main result is the following theorem. Theorem A. Fix a prime p, a connected reductive integral group scheme G, and a pair of prime powers q and q0 both prime to p. Then the following hold, where "'" always means isotypically equivalent. __ ___ (a) Fp(G(q)) ' Fp(G(q0)) if = as subgroups of Zxp. (b) If G is of type An, Dn, or_E6,_and_o is a graph automorphism of G, then Fp(oG(q)) ' Fp(oG(q0)) if = as subgroups of Zxp. (c)If the Weyl group of G contains an element which acts on the maximal torus* * by inverting all elements,_then_Fp(G(q))_' Fp(G(q0)) (or Fp(oG(q)) ' Fp(oG(q0* *)) for o as in (b)) if <- 1, q>= <- 1, q0>as subgroups of Zxp. (d) If G is of type An, Dn for n odd, or E6, and_o_is_a_graph_automorphism of G of order two, then Fp(oG(q)) ' Fp(G(q0)) if <- q>= as subgroups of Zxp. Here, in all cases,_G(q) means the fixed subgroup of the field automorphism__q acting on G(Fq), and oG(q) means the fixed subgroup of o_q acting on G(Fq). ____________ 2000 Mathematics Subject Classification. Primary 20D06. Secondary 55R37, 20D* *20. Key words and phrases. groups of Lie type, fusion systems, classifying space* *s, p-completion. C. Broto is partially supported by MEC grant MTM2007-61545. B. Oliver is partially supported by UMR 7539 of the CNRS. 1 2 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER We remark here that this theorem does not apply when comparing fusion systems of SOn (q) and SOn (q0) for even n, at least not when q or q0 is a power of 2, * *since SOn(K) is not connected when K is algebraically closed of characteristic two. Instead, one must compare the groups n (-). For example, for even n 4, +n(4) and +n(7) have equivalent 3-fusion systems, while SO+n(4) and SO+n(7) do not. Points (a)-(c) of Theorem A will be proven in Proposition 3.2, where we deal with the more general situation where G is reductive (thus including cases such as G = GLn). Point (d) will be proven as Proposition 3.3. In all cases, this wi* *ll be done by showing that the p-completed classifying spaces of the two groups are homotopy equivalent. A theorem of Martino and Priddy (Theorem 1.5 below) then implies that the fusion systems are isotypically equivalent. Since p-completion of spaces plays a central role in our proofs, we give a v* *ery brief outline here of what it means, and refer to the book of Bousfield and Kan [BK ]* * for more details. They define p-completion as a functor from spaces to spaces, whic* *h we denote (-)^phere, and this functor comes with a map X ~p(X)----!X^pwhich is nat* *ural in X. For any map f :X ---! Y , f^pis a homotopy equivalence if and only if f i* *s a mod p equivalence; i.e., H*(f; Fp) is an isomorphism from H*(Y ; Fp) to H*(X; F* *p). A space X is called "p-good" if ~p(X^p) is a homotopy equivalence (equivalently, ~p(X) is a mod p equivalence). In particular, all spaces with finite fundament* *al group are p-good. If X is p-good, then ~p(X): X ---! X^pis universal among all mod p equivalences X ---! Y . If X and Y are both p-good, then X^p' Y ^p(the p-completions are homotopy equivalent) if and only if there is a third space Z,* * and mod p equivalences X ---! Z --- Y . By a theorem of Friedlander (stated as Theorem 3.1 below), B(oG(q))^pis the homotopy fixed space (Definition 2.1) of the action of o_q on BG(C)^p. Theorem A follows from this together with a general result about homotopy fixed spaces (Theorem 2.4), which says that under certain conditions on a space X, two self homotopy equivalences have equivalent homotopy fixed sets if they generate the same closed subgroup of the group of all self equivalences. Corresponding results for the Suzuki and Ree groups can also be shown using this method of proof. But since there are much more elementary proofs of these results (all equivalences are induced by inclusions of groups), and since it se* *emed difficult to find a nice formulation of the theorem which included everything, * *we decided to leave them out of the statement. As another application of these results, we prove that for any prime p and a* *ny prime power q 1 (mod p), the fusion systems Fp(G2(q)) and Fp(3D4(q)) are isotypically equivalent if p 6= 3, and the fusion systems Fp(F4(q)) and Fp(2E6(* *q)) are isotypically equivalent if p 6= 2 (Example 4.5). However, while this provi* *des another example of how our methods can be applied, the first equivalence (at le* *ast) can also be shown by much simpler methods. Theorem A is certainly not surprising to the experts, who are familiar with * *it by observation. It seems likely that it can also be shown directly using a pur* *ely algebraic proof, but the people we have asked do not know of one, and there does not seem to be any in the literature. There is a very closely related result by* * Michael Larsen [GR , Theorem A.12], restated below as Theorem 3.4. It implies that two Chevalley groups G(K) and G(K0) over algebraically closed fields of characteris* *tic prime to p have equivalent p-fusion systems when defined appropriately for these infinite_groups. There are standard methods for comparing the finite subgroups * *of G(Fq) (of order prime to q) with those in its finite Chevalley subgroups (see, * *e.g., EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 3 Proposition 3.5), but we have been unable to get enough control over them to pr* *ove Theorem A using Larsen's theorem. The paper is organized as follows. In Section 1, we give a general survey of* * fusion categories of finite groups and their relationship to p-completed classifying s* *paces. Then, in Section 2, we prove a general theorem (Theorem 2.4) comparing homotopy fixed points of different actions on the same space, and apply it in Section 3 * *to prove Theorem A. In Section 4, we show a second result about homotopy fixed points, which is used to prove the result comparing fusion systems of G2(q) and 3D4(q), and F4(q) and 2E6(q). We finish with a brief sketch in Section 5 of some elemen* *tary techniques for proving special cases of Theorem A for some classical groups, and more generally a comparison of fusion systems of classical groups at odd primes. 1.Fusion categories We begin with a quick summary of what is needed here about fusion systems of finite groups. Definition 1.1. For any finite group G and any prime p, Fp(G) denotes the cat- egory whose objects are the p-subgroups of G, and where Mor Fp(G)(P, Q) = {' 2 Hom (P, Q) | ' = cx for some x 2 G} . Here, cx denotes the conjugation homomorphism: cx(g) = xgx-1. If S 2 Sylp(G) is a Sylow p-subgroup, then FS(G) Fp(G) denotes the full subcategory with objects the subgroups of S. A functor F :C ---! C0 is an equivalence of categories if it induces bijecti* *ons on isomorphism classes of objects and on all morphism sets. This is equivalent to the condition that there be a functor from C0 to C such that both composites are naturally isomorphic to the identity. An inclusion of a full subcategory is* * an equivalence if and only if every object in the larger category is isomorphic to* * some object in the smaller one. Thus when G is finite and S 2 Sylp(G), the inclusion FS(G) Fp(G) is an equivalence of categories by the Sylow theorems. ~= In general, we write : C1 ---! C2 to mean that is an isomorphism of cate- gories (bijective on objects and on morphisms); and : C1 -'--!C2 to mean that is an equivalence of categories. In the following definition, for any finite G, ~G denotes the forgetful func* *tor from Fp(G) to the category of groups. Definition 1.2. Fix a prime p, a pair of finite groups G and G*, and Sylow p- subgroups S 2 Sylp(G) and S* 2 Sylp(G*). ~= (a) An isomorphism ': S ---! S* is fusion preserving if for all P, Q S and ff 2 Hom (P, Q), ff 2 MorFp(G)(P, Q) () 'ff'-1 2 MorFp(G*)('(P ), '(Q)). (b) An equivalence of categories T :Fp(G) ---! Fp(G*) is isotypical if there is a natural isomorphism~of functors ! :~G ---! ~G* O T ; i.e., if there are isomorphisms !P :P --=-!T (P ) such that !Q O ' = T (') O!P for each ' 2 Hom G(P, Q). In other words, in the above situation, an isomorphism ': S ---!~S* is fusion preserving if and only if it induces an isomorphism from FS(G) -=--!FS*(G*) by 4 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER sending P to '(P ) and ff to 'ff'-1. Any such isomorphism of categories extends to an equivalence Fp(G) -'--!Fp(G*), which is easily seen to be isotypical. In * *fact, two fusion categories Fp(G) and Fp(G*) are isotypically equivalent if and only * *if there is a fusion preserving isomorphism between Sylow p-subgroups, as is shown in the following proposition. Proposition 1.3. Fix a pair of finite groups G and G*, a prime p and Sylow p-subgroups S G and S* G*. Then the following are equivalent: ~= (a) There is a fusion preserving isomorphism ': S ---! S*. (b) Fp(G) and Fp(G*) are isotypically equivalent. ~= (c)There are bijections Rep (P, G) ---! Rep (P, G*), for all finite p-groups * *P , which are natural in P . Proof. This was essentially shown by Martino and Priddy [MP ], but not complete* *ly explicitly. By the above remarks, (a) implies (b). (b =) c) : Fix an isotypical equivalence T :Fp(G) ---! Fp(G*), and let ! be an associated natural isomorphism. Thus !P 2 Iso(P, T (P )) for each p-subgroup P G, and !Q Off = T (ff) O!P for all ff 2 Hom G(P, Q). For each p-group Q, ! defines a bijection from Hom (Q, G) to Hom (Q, G*) by sending ae to !ae(Q)Oae. * *For ff, fi 2 Hom (Q, G), the diagram !ff(Q) Q _____ff__//ff(Q)_______//_OTO(ff(Q)) || || flO T(fl)O || O O || fi fflffl!fi(Q) fflffl Q _________//fi(Q)_______//_T (fi(Q)) , together with the fact that T is an equivalence, proves that ff and fi are G-co* *njugate (there exists fl which makes the left hand square commute) if and only if !ff(Q* *)Off and !fi(Q)Ofi are G*-conjugate (there exists T (fl)). Thus T induces bijections ~= : Rep(Q, G) ---! Rep (Q, G*); and similar arguments show that is natural in Q. ~= (c =) a) : Fix a natural bijection : Rep(-, G) -! Rep(-, G*) of functors on finite p-groups. By naturality, preserves kernels, and hence restricts to a b* *ijection between classes of injections. In particular, there are injections of S into G** * and S* into G, and thus S ~=S*. Since conjugation defines a fusion preserving isomorph* *ism between any two Sylow p-subgroups of G*, we can assume S* = Im( S(inclGS)). Using the naturality of , it is straightforward to check that S(inclGS) is fu* *sion preserving as an isomorphism from S to S*. We also note the following, very elementary result about comparing fusion sy* *s- tems. Proposition 1.4. Fix a prime p, and a pair of groups G1 and G2 such that Fp(G1) is isotypically equivalent to Fp(G2). Then the following hold, where "'" always means isotypically equivalent. (a) If Zi Z(Gi) is central of order prime to p, then Fp(Gi=Zi) ' Fp(Gi). (b) If Z1 Z(G1) is a central p-subgroup, and Z2 G2 is its image under some fusion preserving isomorphism between Sylow p-subgroups of the Gi, then Fp(G1) ' Fp(CG2(Z2)) and Fp(G1=Z1) ' Fp(CG2(Z2)=Z2). EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 5 (c)Fp([G1, G1]) ' Fp([G2, G2]). Proof. Points (a) and (b) are elementary. To prove (c), first fix Sylow subgrou* *ps ~= Si 2 Sylp(Gi) and a fusion preserving isomorphism ': S1 ---! S2. By the focal subgroup theorem (cf. [Go , Theorem 7.3.4]), '(S1 \ [G1, G1]) = S2 \ [G2, G2]. * *By [BCGLO2 , Theorem 4.4], for each i = 1, 2, there is a unique fusion subsystem * *"of p-power index" in FSi(Gi) over the focal subgroup Si\ [Gi, Gi], which must be the fusion system of [Gi, Gi]. Hence ' restricts to an isomorphism which is fus* *ion preserving with respect to the commutator subgroups. Proposition 1.4 implies, for example, that whenever Fp(GLn(q)) ' Fp(GLn(q0)) for q and q0prime to p, then there are also equivalences Fp(SLn(q)) ' Fp(SLn(q0* *)), Fp(P SLn(q)) ' Fp(P SLn(q0)), etc. The following theorem of Martino and Priddy shows that the p-fusion in a fin* *ite group is determined by the homotopy type of its p-completed classifying space. * *The converse (the Martino-Priddy conjecture) is also true, but the only known proof uses the classification of finite simple groups [O1 , O2]. Theorem 1.5. Assume p is a prime, and G and G0 are finite groups, such that BG^p' BG0^p. Then Fp(G) and Fp(G0) are isotypically equivalent. Proof. This was shown by Martino and Priddy in [MP ]. The key ingredient in the proof is a theorem of Mislin [Ms , pp.457-458], which says that for any finite * *p-group Q and any finite group G, there is a bijection B^p ^ Rep(Q, G) -----!~=[BQ, BGp] , where B^psends the class of a homomorphism ae: Q ___! G to the p-completion of Bae: BQ ---! BG. Thus any homotopy equivalence BG^p--'-!BG0^pinduces bijections Rep(Q, G) ~=Rep(Q, G0), for all p-groups Q, which are natural in Q. * *The theorem now follows from Proposition 1.3. The following proposition will also be useful. When H G is a pair of group* *s, we regard Fp(H) as a subcategory of Fp(G). Proposition 1.6. If H G is a pair of groups, then Fp(H) is a full subcategory of Fp(G) if and only if the induced map Rep (P, H) -----! Rep(P, G) is injective for all finite p-groups P . Proof. Assume Rep (P, H) injects into Rep (P, G) for all P . For each pair of * *p- subgroups P, Q H and each ' 2 Hom G(P, Q), [inclGP] = [inclGQO'] in Rep(P, G), so [inclHP] = [inclHQO'] in Rep(P, H), and thus ' 2 Hom H(P, Q). This proves th* *at Fp(H) is a full subcategory of Fp(G). Conversely, assume Fp(H) is a full subcategory. Fix a finite p-group P , and ff, fi 2 Hom (P, H) such that [ff] = [fi] in Rep(P, G). Let ' 2 Hom G(ff(P ), f* *i(P )) be such that 'Off = fi. Then ' 2 Hom H(ff(P ), fi(P )) since Fp(H) is a full subca* *tegory, and so [ff] = [fi] in Rep(P, H). This proves injectivity. Our goal in the next three sections is to construct isotypical equivalences * *between fusion systems of finite groups at a prime p by constructing homotopy equivalen* *ces between their p-completed classifying spaces. 6 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER 2.Homotopy fixed points of self homotopy equivalences We start by defining homotopy orbit spaces and homotopy fixed spaces for a s* *elf homotopy equivalence of a space; i.e., for a homotopy action of the group Z. As usual, I denotes the unit interval [0, 1]. Definition 2.1. Fix a space X, and a map ff: X ---! X. (a) When ff is a homeomorphism, the homotopy orbit space Xhffand homotopy fixed space Xhffof ff are defined as follows: o Xhff= (X x I)=~ , where (x, 1) ~ (ff(x), 0) for all x 2 X. o Xhff is the space of all continuous maps fl :I ---! X such that fl(1) = ff(fl(0)). (b) When ff is a homotopy equivalence but not a homeomorphism, define the dou- ble mapping telescope of ff by setting Tel(ff) = (X x I x Z)=~ where (x, 1, n) ~ (ff(x), 0, n + 1) 8 x 2 X, n 2 * *Z. Let bff:Tel(ff) ---! Tel(ff) be the homeomorphism bff([x, t, n]) = [x, t, * *n - 1]. Then set Xhff= Tel(ff)hbff and Xhff= Tel(ff)hbff, where Tel(ff)hbffand Tel(ff)hbffare defined as in (a). The space Xhff, when defined as in (a), is also known as the mapping torus o* *f ff. In this situation, Xhffis clearly the space of sections of the bundle Xhff-pff-* *-!S1, defined by identifying S1 with I=(0 ~ 1). When ff is a homotopy equivalence, the double mapping telescope Tel(ff) is homotopy equivalent to X. Thus the idea in part (b) of the above definition is * *to replace (X, ff) by a pair (Xb, bff) with the same homotopy type, but such that * *bffis a homeomorphism. The following lemma helps to motivate this approach. Lemma 2.2. Fix spaces X and Y , a homotopy equivalence f :X ---! Y , and homeomorphisms ff: X ---! X and fi :Y ---! Y such that fi Of ' f Off. Then there are homotopy equivalences Xhff' Yhfi and Xhff' Y hfi, where these spaces are defined as in Definition 2.1(a). Proof. Fix a homotopy F :X x I ---! Y such that F (x, 0) = f(x) and F (x, 1) = fi-1 Of Off. Define h: Xhff ------! Yhfi =(XxI)=~ =(Y xI)=~ by setting h(x, t) = (F (x, t), t). If g is a homotopy inverse to f and G is a * *homotopy from g to ff-1 Og Ofi, then these define a map from Yhfito Xhffwhich is easily * *seen to be a homotopy inverse to h. We thus have a homotopy equivalence beween Xhffand Xhfiwhich commutes with the projections to S1. Hence the spaces Xhffand Y hfiof sections of these bundles are homotopy equivalent. Lemma 2.2 also shows that when ff is a homeomorphism, the two constructions * *of Xhffand Xhffgiven in parts (a) and (b) of Definition 2.1 are homotopy equivalen* *t. EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 7 Remark 2.3. The homotopy fixed point space Xhffof a homeomorphism ff can also be described as the homotopy pullback of the maps X -----! X x X -(Id,ff)----X , where is the diagonal map (x) = (x, x). In other words, Xhffis the space of triples (x1, x2, OE), where x1, x2 2 X, and OE is a path in XxX from (x1) = (x* *1, x1) to (x2, ff(x2)). Thus OE is a pair of paths in X, one from x1 to x2 and the ot* *her from x1 to ff(x2), and these two paths can be composed to give a single path fr* *om x2 to ff(x2) which passes through the (arbitrary) point x1. Hence this definiti* *on is equivalent to the one given above. It helps to explain the name "homotopy fixed point set", since the ordinary pullback of the above maps can be identified wit* *h the space of all x 2 X such that ff(x) = x. For any space X, we set Hbi(X; Zp) = lim-Hi(X; Z=pk) for each i, and let Hb*(X; Zp) be the sum of the Hbi(X; Zp). If H*(X; Fp) is finite in each degree, then bH*(X; Zp) is isomorphic to the usual cohomology ring H*(X; Zp) with coef- ficients in the p-adics. By the p-adic topology on Out(X), we mean the topology for which {Uk} is a basis of open neighborhoods of the identity, where Uk Out(X) is the group of automorphisms which induce the identity on H*(X; Z=pk). Thus this topology is Hausdorff if and only if Out(X) is detected on bH*(X; Zp). Theorem 2.4. Fix a prime p. Let X be a connected, p-complete space such that o H*(X; Fp) is noetherian, and o Out (X) is detected on bH*(X; Zp). Let ff and fi be self homotopy equivalences of X which generate the same closed subgroup of Out(X) under the p-adic topology. Then Xhff' Xhfi. Proof. Upon replacing X by the double mapping telescope of ff, we can assume that ff is a homeomorphism. By Lemma 2.2, this does not change the homotopy type of Xhffor Xhfi. Let r 1 be the smallest integer prime to p such that the action of ffr on H*(X; Fp) has p-power order._(The_action_of ff on the noetherian ring H*(X; Fp) has finite order.) Since = , H*(ff; Fp) and H*(fi; Fp) generate the sa* *me subgroup in Aut (H*(X; Fp)), hence have the same order, and so r is also the smallest integer prime to p such that the action of fir on H*(X; Fp) has p-power order. Let pff:Xhff-----! S1 and pfi:Xhfi-----! S1 be the canonical fibrations. Let epff:eXhff-----!S1 and epfi:eXhfi-----!S1 be their r-fold cyclic covers, considered as equivariant maps between spaces wi* *th Z=r-action. Thus eXff' Xhffr(and epffis its canonical fibration), and the same * *for Xefi. Also, since each section of pfflifts to a unique equivariant section of e* *pff, and each equivariant section factors through as section of pffby taking the orbit m* *ap, Xhffis the space of all equivariant sections of epff. Since ffr acts on H*(X; Fp) with order a power of p, this action is nilpoten* *t; and by [BK , II.5.1], the homotopy fiber of epff^phas the homotopy type of X^p'* * X. 8 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER Thus the rows in the following diagram are (homotopy) fibration sequences: Xw ________! Xehff________epff!S1 ww | ww #~p| ~p| # epff^p 1^ X _______! (Xehff)^p____! S p , and so the right hand square is a homotopy pullback. By the definition of p- completion in [BK ], the induced actions of Z=r on S1^pand on (Xehff)^pare free, since the actions on the uncompleted spaces are free. Hence Xhffcan be described (up to homotopy), not only as the space of Z=r-equivariant sections of epff, bu* *t also as the space of Z=r-equivariant liftings of ~p(S1): S1 ---! S1^palong epff^p. I* *n other words, 1 ^ epff^pO- 1 1^ Xhff' fibermapZ=r(S , (Xehff)p) ------! map Z=r(S , S p) (1) (the fiber over ~p(S1)). Consider again the p-completed fibration sequence X ------! (Xehff)^p---epff---!BS1^p, and its orbit fibration X ------! (Xehff)^p=(Z=r) ---bpff---!BS1^p=(Z=r) . Here, ss1(BS1^p) ~= Zp, and ss1(BS1^p=(Z=r)) ~= Zp x Z=r (the completion of Z with respect to the ideals rpiZ). Since ffr acts on H*(X; Z=p) with order a pow* *er of p, it also acts on each H*(X; Z=pk) with order a power of p, and hence the homotopy action of ss1(BS1^p) on X has as image the p-adic closure of . Th* *us the homotopy action of ss1(BS1^p=(Z=r)) on X, defined by the fibration bpff, ha* *s as image the p-adic closure of . ___ b Since fi 2 by assumption, we can represent it by a map S1 ---! S1^p=(Z=r* *). Define Y to be the homotopy pullback defined by the following diagram: 0 Xw_____________! Y _____________p!S1 ww | ww #f| |b # X ________! (Xehff)^p=(Z=r)__bpff!S1^p=(Z=r) . Thus the canonical generator of ss1(S1) induces fi 2 Out (X), and so (Y, p0) ' (Xhfi, pfi). Upon taking r-fold covers and then completing the first row, this * *induces a map of fibrations Xw_______! (Xehfi)^p____epfi!S1^p ww ww '#f| |eb # X _______! (Xehff)^p____epff!S1^p, ___ which is an equivalence since ebis an equivalence (since is dense in )* *. Also, f and ebare equivariant with respect to some automorphism of Z=r. EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 9 The maps S1 --! S1^p-- (X)hZp determine a commutative diagram map Z=r(S1^p, (Xehff)^p)_!mapZ=r(S1, (Xehff)^p) #t| #|u (2) mapZ=r(S1^p, S1^p)____! mapZ=r(S1, S1^p) which in turn induces a map between the respective fibres. The horizontal arrow* *s in (2) are homotopy equivalences because the target spaces in the respective mappi* *ng spaces are p-complete and Z=r acts freely on the source spaces. Hence the fibers of the vertical maps in (2)are homotopy equivalent. Since u has fiber Xhffby (1* *), this proves that Xhffhas the homotopy type of the space of equivariant sections epff^p of the bundle (Xhff)^p---! S1^p. Since this bundle is equivariantly equivalent* * to the one with total space (Xehfi)^p, the same argument applied to fi proves that Xhff' Xhfi. Our main application of Theorem 2.4 is to the case where X = BG^pfor a compact connected Lie group G. Corollary 2.5. Let G be a compact connected Lie group, and let ff, fi 2 Out(BG^* *p) be two self equivalences of the p-completed classifying space. If ff and fi gen* *erate the same closed subgroup of Out(BG^p), then (BG^p)hff' (BG^p)hfi. Proof. From the spectral sequence for the fibration U(n)=G ---! BG ---! BU(n) for any embedding G U(n), we see that H*(BG; Fp) is noetherian. By [JMO , Theorem 2.5], Out(BG^p) is detected by its restriction to BT ^pfor a maximal to* *rus T , and hence by invariant theory is detected by Q ZHb*(BG; Zp). So the hypothe* *ses of Theorem 2.4 hold when X = BG^p. The hypotheses on X in Theorem 2.4 also apply whenever X is the classifying space of a connected p-compact group. The condition on cohomology holds by [DW * * , Theorem 2.3]. Automorphisms are detected by restriction to the maximal torus by [AGMV , Theorem 1.1] (when p is odd) and [Ml , Theorem 1.1] (when p = 2). 3.Finite groups of Lie type We first fix our terminology. Let G be a connected reductive integral group scheme. Thus for each algebraically closed field K, G(K) is a complex connected algebraic group such that for some finite central subgroup Z Z(G(K)), G(K)=Z is the product of a K-torus and a semisimple group. For any prime power q, we let G(q) be the fixed subgroup of the field automorphism _q. Also, if o is any automorphism of G of_finite order, then oG(q) will denote the fixed subgroup of* * the composite o_q on G(Fq). Note that with this definition, when G = P SLn, G(q) does not mean P SLn(q) * *in the usual sense, but rather its extension by diagonal automorphisms (i.e., P GL* *n(q)). By Proposition 1.4, however, any equivalence between fusion systems over groups SLn(-) will also induce an equivalence between fusion systems over P_SLn(-). Also, we are not including the case G = SOn for n even, since SOn(F2) is not co* *n- nected. Instead, when working with orthogonal groups in even dimension, we take G = n (and n(K) = SOn(K) when K is algebraically closed of characteristic different from two). 10 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER The results in this section are based on Corollary 2.5, together with the fo* *llowing theorem of Friedlander. Following the_terminology of [GLS3 ], we define a Stein* *berg endomorphism of an algebraic_group_G over an algebraically closed field to be an algebraic endomorphism _ :G ---! G which is bijective, and whose fixed subgroup is finite. For any connected complex Lie group G(C) with maximal torus T(C), any prime p, and any m 2 Z prime to p, m :BG(C)^p---! BG(C)^pdenotes a self equivalence whose restriction to BT(C)^pis induced by (x 7! xm ) (an "unstable Adams operation"). Such a map is unique up to homotopy by [JMO , Theorem 2.5] (applied to BG ' BG(C), where G is a maximal compact subgroup of G(C)). Theorem 3.1. Fix a connected reductive group scheme G, a prime power q, and a_ prime p which does not divide q. Then for any Steinberg endomorphism _ of G(Fq) with fixed subgroup H, BH^p' (BG(C)^p)h _ for some :_BG(C)^p--'-!BG(C)^p. If _ = o(Fq) OG(_q), where o 2 Aut(G) and _q 2 Aut(Fq) is the automorphism (x 7! xq), then ' Bo(C) O q where q is as described above. Proof. By [Fr, Theorem 12.2], BH^pis homotopy equivalent to a homotopy pullback of maps _ (Id,Id) BG(C)^p--(Id,-)----!BG(C)^px BG(Fq)^p------- BG(C)^p for some ; and hence is homotopy equivalent to (BG(C)^p)h by Remark 2.3. From the proof of Friedlander's theorem, one sees that_ is induced by B_, together * *with the homotopy equivalence BG(C)^p' holim (BG(Fq)et)^p of [Fr, Proposition 8.8]. This equivalence is natural_with respect to the inclusion of a maximal torus T * *in G. Hence when _ = o(Fq) OG(_q), restricts to the action on BT(C)^pinduced by o and (x 7! xq). Theorem 3.1 can now be combined with Corollary 2.5 to prove Theorem A; i.e., to compare fusion systems over different Chevalley groups associated to the same connected group scheme G. This will be done in the next two propositions. Proposition 3.2. Fix a prime p, a connected reductive integral group scheme G, and an automorphism o of G of finite order k. Assume, for each m prime to k, that om is conjugate to o in the group of all automorphisms of G. Let q and q0* * be prime powers prime to p. Assume either __ ___ (a) = as subgroups of Zxp; or (b) there is some _-1 in_the_Weyl_group_of G which inverts all elements of the maximal torus, and <- 1, q>= <- 1, q0>as subgroups of Zxp. Then there is an isotypical equivalence Fp(oG(q)) ' Fp(oG(q0)). Proof. By [JMO , Theorem 2.5], the group Out(BG(C)^p) is detected by restrict- ing maps to a maximal torus. (Every class in this group is represented by0some map which sends BT ^pto itself for some maximal torus T .) Hence q, q 2 Out (BG(C)^p) (the maps whose restrictions to the maximal torus are induced by (x 7! xq) and (x 7! xq0),_respectively)_generate the same closed subgroup of Out (BG(C)^p) whenever = . EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 11 _ _ Since o is an automorphism of G, its actions on G(Fq) and G(Fq0) commute with the field automorphisms _q and _q0. Thus (o_q)k = _qk has finite0fixed subgroup, so o_q also has finite fixed subgroup, and similarly for o_q . So by Theorem 3.* *1, q) o 0 ^ ^ h(BoO q0) B(oG(q))^p' (BG(C)^p)h(BoO and B( G(q ))p ' (BG(C)p) . __ ___ __Assume__= . Then for some m prime to k = |o|, q (q0)m modulo = . Hence Bom O q and Bo O q0generate the same closed subgroup of Out(G(C)^p)_under the p-adic topology, since they generate the same subgroup modulo < qk>. Thus q0) ^ h(Bom O q) ^ h(BoO q) (BG(C)^p)h(BoO ' (BG(C)p) ' (BG(C)p) , where the first equivalence holds by Corollary 2.5, and the second since o and * *om are conjugate in the group of all automorphisms of G. So B(oG(q))^p' B(oG(q0))^* *p, and there is an isotypical equivalence between the fusion systems of these grou* *ps by Theorem 1.5. If - Idis in the Weyl group, then we can regard this as an inner automorphis* *m of G(C) which inverts all elements in a maximal torus. Thus by_[JMO_ ] again,_ -1 * *' Id in this case. So by the same argument0as that just given, <- 1, q>= <- 1, q0>im* *plies (BG(C)^p)h(BoO q)' (BG(C)^p)h(BoO q ), and hence Fp(oG(q)) ' Fp(oG(q0)). __ ___ To make the condition = more concrete, note that for any prime p, and any q, q0 prime to p of order s and s0in Fxp, respectively, ( __ ___ s = s0and vp(qs - 1) = vp(q0s- 1) if p is odd = () q q0 (mod 8) and vp(q2 - 1) = vp(q02-i1)f p = 2 . Corollary 2.5 can also be applied to compare fusion systems of Steinberg gro* *ups with those of related Chevalley groups. Proposition_3.3._Fix a prime p, and a pair q, q0 of prime powers prime to p such that <- q>= as subgroups of Zxp. Then there are isotypical equivalences: (a) Fp(SUn(q)) ' Fp(SLn(q0)) for all n. (b) Fp(Spin-2n(q)) ' Fp(Spin+2n(q0)) for all odd n. (c)Fp(2E6(q)) ' Fp(E6(q0)). Proof. Set G = SLn, Spin2nfor n odd, or the simply connected E6; and let o be t* *he graph automorphism of order two. In all of these cases, o acts by inverting-the* *qele- ments of some maximal torus. Hence by Theorem 3.1, B(oG(q))^p' (BG(C)^p)h 0 and BG(q0)^p' (BG^p)h q . So BG(q)^p'_BG(q0)^pby_Corollary 2.5, [JMO , The- orem 2.5], and the assumption <- q>= ; and there is an isotypical equivalen* *ce between the fusion systems of these groups by Theorem 1.5. Upon combining this with Proposition 1.4, one gets similar results for P SUn* *(q), -2n(q), P -2n(q), etc. This finishes the proof of Theorem A. We now finish the section with some remarks about a possible algebraic proof of this result. The following theorem * *of Michael Larsen in his appendix to [GR ] implies roughly that two Chevalley grou* *ps G(K) and G(K0) over algebraically closed fields K and K0 have equivalent p-fusi* *on 12 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER systems (appropriately defined) for p different from the characteristics of K a* *nd K0. Theorem 3.4. Fix a connected group scheme G, and let K and K0 be two alge- braically closed fields. Then there are bijections ~= 0 P :Rep (P, G(K)) ------! Rep(P, G(K )), for all finite groups P of order prime to char(K) and char(K0), and which are natural with respect to P , and also with respect to automorphisms of G. Proof. Except for the statement of naturality, this is [GR , Theorem A.12]. For fields of the same characteristic, the bijection is induced by the inclusions_o* *f the algebraic closures of their prime subfields ([GR , Lemma A.11]). When K = Fq for a prime q, W (K) is its ring of Witt vectors (the extension of Zq by all ro* *ots of unity of order prime to q), and K0 is the algebraic closure of W (K), then t* *he bijections Rep(P, G(K)) ~=Rep(P, G(W (K))) ~=Rep(P, G(K0)) are induced by the projection W (K) --i K and inclusion W (K) K0 in the obvious way. All of these are natural in P , and also commute with automorphisms of G. _ The next proposition describes how to compare Rep(P, G(Fq)) to the Chevalley and Steinberg groups over Fq. __ _ Proposition 3.5. Fix a connected_algebraic group G over Fq for some q, and oe be a Steinberg endomorphism of G , and set G = C_G(oe). Let P be a finite group, a* *nd consider the map of sets __ ae: Rep(P, G) ------! Rep(P, G). __ __ Fix ' 2 Hom (P, G), and let ['] be its class in Rep(P,_G)._Then ['] 2 Im(ae) if* * and only if ['] is fixed under the action of oe on Rep(P, G). When '(P ) G, set C = C_G(P ), N = N_G(P ), bC= ss0(C), __ and let g 2 N act on C by sending x to gxoe(g)-1. Then there is a bijection ~= B :ae-1([']) ------! bC=C, __ where B([cy O']) = y-1oe(y) for any y 2 G such that y'(P )y-1 G. Also, Aut G('(P )) is the stabilizer, under the action of AutG_('(P )) ~=N=C, of the * *class of the identity element in bC. Proof. This is a special case of_[GLS3 , Theorem_2.1.5], when applied to the set of all homomorphisms '0:P ---! G which are G-conjugate to '. We assumed at first that an algebraic proof of Theorem A could easily be con- structed by applying Theorem 3.4 and Proposition 3.5. But so far, we have been unable to do so, nor do we know of any other algebraic proof of this result. 4. Homotopy fixed points of proxy actions To get more results of this type, we need to look at more general types of a* *ctions and their homotopy fixed points. The concept of "proxy actions" is due to Dwyer and Wilkerson. EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 13 Definition 4.1. For any discrete group G and any space X, a proxy action of G on X is a fibration f :XhG ---! BG with fiber X. An equivalence of proxy actions f :XhG ---! BG on X and f0: YhG ---! BG on Y is a homotopy equivalence ff: XhG ---! YhG such that f0Off = f. The homotopy fixed space XhG of a proxy action f :XhG ---! BG on X is the space of sections s: BG ---! XhG of the fibration f. Any (genuine) action of G on X can be regarded as a proxy action via the Bor* *el construction: XhG = EG xG X is the orbit space of the diagonal G-action on EG x X. In this case, we can identify XhG = map G(EG, X) via covering space theory. If ff: X ---! X is a homeomorphism, regarded as a Z-action, then the mapping torus ffi Xhff= X x I (x, 1) ~ (ff(x), 0) , as defined in Definition 2.1, is homeomorphic to the Borel construction EZ xZ X. So in this case, the homotopy fixed set XhZ of Definition 4.1 is the same as the space Xhffof Definition 2.1. If ff is a self homotopy equivalence of X, then th* *e map from the mapping torus to S1 = BZ need not be a fibration, which is why we need to first replace X by the double telescope before defining the homotopy fixed s* *et. Now assume, furthermore, that X is p-complete, and that the action of ff on Hn(X; Fp) is nilpotent for each n. Then by [BK , Lemma II.5.1], the homotopy fiber of the p-completion (Xhff)^p---! S1^phas the homotopy type of X^p' X. Also, S1^p' BZp, and so this defines a proxy action of the p-adics on X. By the arguments used in the proof of Theorem 2.4, the homotopy fixed space of this action has the homotopy type of Xhff. Some of the basic properties of proxy actions and their homotopy fixed spaces are listed in the following proposition. Proposition 4.2. Fix a proxy action XhG --f-!BG of a discrete group G on a space X. 0 (a) If YhG --f-!BG is a proxy action of the same group G on another space Y , and ': XhG ---! YhG is a homotopy equivalence such that f0 O' = f, then ' induces a homotopy equivalence XhG ' Y hG. (b) Let eXbe the pullback of XhG and EG over BG. Then G has a genuine action on eX, which as a proxy action, is equivalent to that on X. Proof. Point (a) follows easily from the definition. In the situation of (b), the action of G on EG induces a free action on Xe. Consider the two G-maps pr1, efOpr2:EG xG Xe------! EG , where ef:eX---! EG is the map coming from the pullback square used to define Xe. By the universality of EG, all maps from EG xG Xe to EG are equivariantly homotopic (cf. [Hu , Theorem 4.12.4], and recall that these spaces are the total spaces of principal G-bundles). So upon passing to the orbit space, the composi* *te fe=G EG xG Xe-pr2=G----!XhG -----!=fBG 14 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER is homotopic to pr1=G, the map which defines the proxy action of G induced by the action on Xe. By the homotopy lifting property for the fibration f, pr2=G is homotopic to a map ff such that f Off = pr1=G, and this is an equivalence betwe* *en the proxy actions. If XhG --f-!BG is a proxy action of G on X, and H G is a subgroup, then the pullback of BH and XhG over BG defines a proxy action XhH -f|H--!BH of H on X. The first statement in the following proposition is due to Dwyer and Wilkerson [DW , 10.5]. Proposition 4.3. Let f :XhG ---! BG be a proxy action of G on X, and let H be a normal subgroup of G with quotient group ss = G=H. Then the following hold. (a) There is a proxy action of ss on XhH with XhG ' (XhH )hss. (b) Assume G0 G and H0 = H \ G are such that the inclusion induces an ~= isomorphism G0=H0 ---! G=H = ss. Assume also that the natural map XhH ---! XhH0 induced by restricting the action of G is a homotopy equiv- alence. Then XhG ---! XhG0 is also a homotopy equivalence. Proof. We give an argument for (a) that will be useful in the proof of (b). Wri* *te BbH = EG=H ' BH and bBG = EG xG Ess ' BG, where EG xG Ess is the orbit space of the diagonal action of G on EG x Ess. Let fl :BG --'---!bBG and Bb': bBH -----! bBG be induced by the diagonal map EG ---! EG x Ess and its composite with the inclusion E': EH ---! EG. The adjoint of the projection bBH x Ess = EG xH Ess ------! EG xG Ess = bBG is a ss-equivariant map Ess ---! map (BbH, bBG)Bb', where map (BbH, bBG)Bb'is t* *he space of all maps homotopic to Bb'. Consider the following homotopy pullback diagram of spaces with ss-action: Y ______! map (EG=H, XhG)[Bb'] | | |f0O- # # Ess_______! map (EG=H, bBG)Bb'. Here, f0 = fl Of :XhG ---! BbG, and map (-, -)[Bb']means the space of all maps whose composite with f0 is homotopic to bB'. Since Ess is contractible, Y is t* *he homotopy fiber of the map on the right, and hence homotopy equivalent to XhH . After taking homotopy fixed spaces (-)hss, and since (EG=H x Ess)=G ~=bBG, we get a new homotopy pullback square (XhH )hss'Y hss____! map (BbG, XhG)cbB' | | # # * 'Esshss_____! map (BbG, bBG)Bb'. Thus (XhH )hssis the homotopy fiber of the right hand map, hence equivalent to the space of sections of the fibration XhG ---! BG, which is XhG . Now assume that we are in the situation of (b). In this situation, if G acts* * on X, the restriction map XhH --'-! XhH0 is ss-equivariant, provided we use the models for XhH and XhH0 constructed above. Taking homotopy fixed points on both EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 15 sides for the action of ss, we see that the inclusion XhG ---! XhG0 is a homoto* *py equivalence. Alternatively, by Proposition 4.2(a,b), it suffices to prove this for a genu* *ine action of G on X. In this case, XhG = map G(EG, X) ' map G(EG x Ess, X) ~=map ss(Ess, mapH (EG, X)) , where the last equivalence follows by adjunction. We can identify map H(EG, X) with XhH , and thus XhG ' (XhH )hss. In the situation of (b), we get a commutat* *ive square XhG ' map G(EG x Ess, X)___! mapss(Ess, mapH (EG, X)) ' (XhH )hss #r1| #r2| XhG0 ' map G0(EG x Ess, X)__! map ss(Ess, mapH0(EG, X)) ' (XhH0)hss. where r1 and r2 are induced by restriction to G0 or H0. Since r2 is a homo- topy equivalence by assumption (and by Proposition 4.2(a)), r1 is also a homoto* *py equivalence. The following theorem deals with homotopy fixed points of actions of K x Z, where K is a finite cyclic group of order prime to p. This can be applied when K is a group of graph automorphisms of BG^p(and G is a compact connected Lie group), or when K is a group of elements of finite order in Zxp(for odd p). Theorem 4.4. Fix a prime p. Let X be a connected, p-complete space such that o H*(X; Fp) is noetherian, and o Out (X) is detected on bH*(X; Zp). Fix a finite cyclic group K = of order r prime to p, together with a proxy action f :XhK ---! BK of K on X. Let fi :XhK ---! XhK be a self homotopy equivalence such that f Ofi = f; and set ff = fi|X , a self homotopy equivalenc* *e of X. Assume H*(ff; Fp) is an automorphism of H*(X; Fp) of p-power order (equiva- lently, the action is nilpotent). Let ~: X ---! X be the self homotopy equivale* *nce induced by lifting a loop representing g 2 ss1(BK) to a homotopy of the inclusi* *on X ---! XhK . Then Xh(~ff)' (XhK )hff. Proof. Upon first replacing X by the pullback of XhK and EK over BK, and then by taking the double mapping telescope of the map from that space to itself induced by fi, we can assume that X has a genuine free action of the group K x * *Z, and that ~ and ff are (commuting) homeomorphisms of X which are the actions of generators of K and of Z. In particular, (~ff)r = ffr. For each k 1, ff acts on H*(X; Z=pk) as an automorphism of p-power order. Since r is prime to p, this implies that ff and ffr generate thersame closed su* *bgroup of Out(X). So by Theorem 2.4, the inclusion of Xhffinto Xhff is a homotopy equivalence. By Proposition 4.3(b), applied with G = <~> x , G0 = <~ff>, and H = , the inclusion of Xh(Kx)into Xh(~ff)is a homotopy equivalence. By Proposition 4.3(a), Xh(Kx)' (XhK )hff, and this proves the theorem. The following result comparing fusion systems of G2(q) and 3D4(q), and those of F4(q) and 2E6(q), is well known. For example, the first part follows easily * *from the lists of maximal subgroups of these groups in [Kl1] and [Kl2], and also fol* *lows 16 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER from the cohomology calculations in [FM ] and [Mi ] (together with Theorem 1.5). We present it here as one example of how Theorem 4.4 can be applied. Example 4.5. Fix a prime p, and a prime power q 1 (mod p). Then the following hold. (a) If p 6= 3, the fusion systems Fp(G2(q)) and Fp(3D4(q)) are isotypically eq* *uiv- alent. (b) If p 6= 2, the fusion systems Fp(F4(q)) and Fp(2E6(q)) are isotypically eq* *uiv- alent. Proof. To prove (a), we apply Theorem 4.4, with X = BSpin8(C)^p' BSpin(8)^p, with K ~=C3 having the action on X induced by the triality automorphism, and with ff = q the unstable Adams operation on X. We first show that the inclusion of G2 into Spin(8) induces a homotopy equiv- alence (BG2)^p' XhK . Since there is always a map from the fixed point set of an action to its homotopy fixed point set, the inclusion of G2(C) ~=Spin8(C)K (cf. [GLS3 , Theorem 1.15.2]) into Spin8(C) induces maps (BG2)^p---! XhK ---! X. The first map is a monomorphism in the sense of Dwyer and Wilkerson [DW , x3.2* *], since the composite is a monomorphism. By [BM , Theorem B(2)], XhK is the classifying space of a connected 2-compact group. Hence by [BM , Theorem B(2)], H*(XhK ; Qp) is the polynomial algebra generated by the coinvariants QH*(X; Qp)K ; i.e, the coinvariants of the K-acti* *on on the polynomial generators of H*(Spin8(C); Qp). For any compact connected Lie group G with maximal torus T , H*(BG; Q) is the ring of invariants of the actio* *n of the Weyl group on H*(BT ; Q) [Bor, Proposition 27.1], and is a polynomial algeb* *ra with degrees listed in [ST , Table VII]. In particular, H*(X; Qp) has polynomial generators are in degrees 4, 8, 12, 8, and an explicit computation shows that K* * fixes generators in degrees 4 and 12. Thus H*(XhK ; Qp) ~=H*(BG2(C); Qp) (as graded Qp-algebras). It follows from [MN , Proposition 3.7] that (BG2)^p---! XhK is * *an isomorphism of connected 2-compact groups because it is a monomorphism and a rational isomorphism. Now let ~ 2 Aut(X) generate the action of K. By Theorem 3.1, Xh(~ff)' B(3D4(q))^p and (XhK )hff' (BG2^p)hff' BG2(q)^p. Since q 1 (mod p), the action of ff = q on H*(X; Fp) has p-power order. Hence B(3D4(q))^p' BG2(q)^pby Theorem 4.4, and so these groups have isotypically equivalent p-fusion systems by Theorem 1.5. This proves (a). Now set X = BE6(C)^pand K = , where o is an outer automorphism of order two. For each k 0, o acts on H2k(X; Qp) via (-1)k: this follows since H*(X; Qp) injects into the cohomology of any maximal torus and o acts on an appropriate choice of maximal torus by via (g 7! g-1). Since H*(X; Qp) is poly- nomial with generators in degrees 4, 10, 12, 16, 18, 24, [BM , Theorem B(2)] im* *plies that H*(XhK ; Qp) is polynomial with generators in degrees 4, 12, 16, 24, and h* *ence is isomorphic to H*(BF4(C); Qp). The rest of the proof of (b) is identical to t* *hat of (a). 5. Classical groups In the case of many of the classical groups, there is a much more elementary approach to Theorem A. Recall that the modular character OV of an Fq[G]-module EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 17 V is defined by identifying Fxqwith a subgroup of Cx , and then letting OV (g) * *2 C (when (|g|, q) = 1) be the sum of the eigenvalues of V g-!V lifted to C. We always consider this in the case where G has order prime to q, and hence when two representations with the same character are isomorphic. See [Se, x18] for m* *ore details. For any finite group G, let Rep n(G) be the set of isomorphism classes of n- dimensional irreducible complex representations (i.e., Repn(G) = Rep(G,_GLn(C))_ in the notation used elsewhere). For any prime p and any q prime to p, (bZ* *p)x denotes the closure of the subgroup generated by q. In the following theorem, we set GL+n(q) = GLn(q) and GL-n(q) = GUn(q) for convenience. Proposition 5.1. Fix a prime p, and let q be a prime power which is prime to p. (a) Fix n 2 and ffl = 1. For any finite p-group P , Rep (P, GLffln(q)) can* * be identified with the set of those V 2 Repn(P ) such that OV (gfflq) = OV (g* *) for all g 2 P . (b) Assume p is odd and fix n 1. G = Sp2n(q) and G1 = GO2n+1(q). Then for any finite p-group P , Rep(P, Sp2n(q)) and Rep(P, GO2n+1(q)) can be identified with the set of those V 2 Rep 2n(P ) ~= Rep2n+1(P ) such th* *at OV (gq) = OV (g) = OV (g-1) for all g 2 P . In particular, the fusion syst* *ems Fp(Sp2n(q)) and Fp(GO2n+1(q)) are isotypically equivalent. Proof. Let K C be the subfield generated by all p-th power roots of unity. For each r 2 Zxp, let _r 2 Aut(K) be the field automorphism _r(i) = ir for each root of unity i. (a) Let bKbe the extension of Qq by all roots of unity prime to q,_let A Qq b* *e the ring of integers, and let p A be the maximal ideal. Thus A=p ~=Fq. By modular representation theory (cf. [Se, Theorems 33 & 42]), for each finite p-group P ,* * there ~= is an isomorphism of representation rings RKb(P ) ----! R_Fq(P ), which sends t* *he class of a bK[P ]-module V to M=pM for any P -invariant A-lattice M V . This clearly sends an actual representation to an actual representation. If M1 V1 and M2 V2 are such that M1=pM1 and M2=pM2_have an irreducible factor in common, then since |P | is invertible in Fq and in A, any nonzero homomorphism ' 2 Hom P(M1=pM1, M2=pM2) can be lifted (by averaging over the elements of P ) to a homomorphism b'2 Hom P(V1, V2). From this we see that the isomorphism RKb(P ) ~=R_Fq(P ) restricts to a bijection between irreducible representations* *, and also between n-dimensional representations for any given n. So _ Rep(P, GLn(Fq)) ~=Rep(P, GLn(Kb)) ~=Rep(P, GLn(K)) ~=Rep(P, GLn(C)), where the last two bijections follow from [Se, Theorem 24]. _ The centralizer of any finite p-subgroup of GLn(Fq) is a product of general * *linear groups, and hence_connected. Thus by Proposition 3.5, Rep(P, GLffln(q)) injects* * into Rep (P, GLn(Fq)), and its image is the_set of representations which are fixed b* *y the Steinberg endomorphism _fflqon GLn(Fq). _ _ Fix V 2 Rep(P, GLn(Fq)) and g 2 P , let ,1, . .,.,n 2 Fq be the eigenvalues * *of the action of g on V , and let i1, . .,.in 2 K be the corresponding p-th power * *roots of unity in C. Then _q(V ) has eigenvalues ,q1, . .,.,qn, _-1(V ) = V *has eige* *nvalues 18 CARLES BROTO, JESPER M. MOLLER, AND BOB OLIVER ,-11, . .,.,-1n, and so _fflq(V ) has eigenvalues ,fflq1, . .,.,fflqn. This pro* *ves that O_fflq(V()g) = Offlq1+ . .+.Offlqn= OV (gfflq) for all V 2 Rep n(P ) and all g 2 P . Thus V 2 Rep n(P ) is in the image of Rep (P, GLffln(q)) if and only if OV (gfflq) = OV (g) for all g 2 P . (b)_ Now assume p is odd, and let P be a finitePp-group. For any irreducible F q[P ]-representation W which is self dual, g2P OW (g2) 6= 0: this is shown* * in [BtD , Proposition II.6.8] for complex representations,Pand the same proof appl* *ies in our situation. Since |P | is odd, this implies that g2P OW (g) 6= 0, and h* *ence that W is the trivial representation. In other words, the_only self dual irredu* *cible representation is the trivial one. Hence every self dual Fq[P ]-representation * *V has the form V = V0 W W 0, where P acts trivially on V0 and has no fixed component on W and W 0, W 0~=W *, and Hom _Fq[P](W, W 0) = 0. _ Fix a self dual Fq[P ]-representation V , and write V = V0 W W 0as above. Fix ffl = 1, where ffl = +1 if dimF_q(V ) is odd, and write "ffl-symmetric" to* * mean symmetric (ffl = +1) or_symplectic (ffl = -1). For any nondegenerate ffl-symmet* *ric ~= form b0 on V0 and any Fq[P ]-linear isomorphism f :W 0---! W *, there is a non- degenerate ffl-symmetric form b on V defined by b((v1, w1, w01), (v2, w2, w02)) = b0(v1, v2) + f(w01)(w2) + fflf(w02)(w1* *)(3) for vi2 V0, wi2 W , and w0i2 W 0. Conversely, if b is any nonsingular ffl-symme* *tric form on V , then b must be nonsingular on V , and zero on W and on W 0, and hence has the form (3)for some_b0 and f. Since all such forms are isomorphic, this proves that Rep(P, Sp2n(Fq)) can be_identified with the set of self dual e* *le- ments of Rep2n(P ), and Rep(P, GO2n+1(Fq)) with the set of self dual elements of Rep 2n+1(P ). _ Thus Rep(P, Sp2n(Fq)) Rep2n(P ) and Rep(P, GO2n+1(q))_ Rep2n+1(P_) are both the sets of_self dual elements. _Since GO2n+1(Fq) = SO2n+1(Fq) x < Id>, Rep (P, SO2n+1(Fq)) = Rep(P, GO2n+1(Fq)). Also, as we just saw, each odd dimen- sional self dual_P -representation has_odd dimensional fixed component, and thus Rep (P, SO2n+1(Fq)) = Rep(P, Sp2n(Fq)) as subsets of Rep2n+1(P ). _ * * _ We claim that the centralizer of any finite p-subgroup of Sp2n(Fq) or SO2n+1* *(Fq) is connected. To see this, fix such a subgroup P , let V be the corresponding representation with symmetric or symplectic form b, and let V = V0 W W 0be a decomposition such that b is as in (3). Then the centralizer of P in Aut(V,_b* *) is the product_of Aut(V0, b0) with Aut(W ), and hence its centralizer in Sp2n(Fq) * *or SO2n+1(Fq) is connected. We now apply Proposition 3.5, exactly as in the proof of (a), to show that f* *or a p- group P , Rep(P, Sp2n(q)) injects into Rep2n(P ) with image the set of those V * *with OV (g) = OV (gq) = OV (g-1) for all g 2 P ; and similarly for Rep(P, SO2n+1(q)). For the linear and unitary groups, Theorem A follows immediately from Propo- sition 5.1(a). Also, Proposition 5.1(b) implies that when p is odd, Theorem A h* *olds for the symplectic and odd orthogonal groups; and also that Fp(Sp2n(q)) ' Fp(GO2n+1(q)) ' Fp(SO2n+1(q)) for each odd p, each n 1, and each q prime to p. Theorem A for the even orthogonal groups then follows from the following observation. EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE 19 Proposition 5.2. For each odd prime p, each prime power q prime to p, and each n 1, Fp(GO2n(q)) ' Fp(SO2n+1(q)) ' Fp(Sp2n(q)) if qn 6 1 (mod p) Fp(GO2n(q)) ' Fp(SO2n-1(q)) ' Fp(Sp2n-2(q)) if qn 6 1 (mod p) . Proof. Any inclusion GOk (q) GOk+1(q) induces an injection of Rep(P, GOk (q)) into Rep(P, GOk+1(q)) for each p-group P . Thus Fp(GOk (q)) is a full subcatego* *ry of Fp(GOk+1(q)) by Proposition 1.6. Hence we get an equivalence of p-fusion categories whenever GOk (q) has index prime to p in GOk+1(q). By the standard formulas for the orders of these groups, [GO2n+1(q) : GO2n(q)]= qn(qn 1) [GO2n(q) : GO2n-1(q)]= qn-1(qn 1) , and the proposition follows. Another consequence of Proposition 5.1 is the following: Proposition 5.3. Fix an odd prime p, and a prime power q prime to p. Set s = order(q) mod p. (a) If s is even, then for each n 1, the inclusion Sp2n(q) GL2n(q) induces* * an equivalence Fp(Sp2n(q)) ' Fp(GL2n(q)) of fusion systems. (b) If s 2 (mod 4), then for each n 1, Sp2n(q) Sp2n(q2) induces an equivalence of p-fusion systems. Proof. If s is even, then -1 is a power of q modulo p, and also modulo pn for all n 2. Hence if P is a p-group, and V 2 Rep 2n(P ) is such that OV (g) = OV (gq) for all g 2 P , then also OV (g) = OV (g-1) for all g. So by Propositio* *n 5.1, Rep (P, Sp2n(q)) ~=Rep(P, GL2n(q)), and so (a) follows from Proposition 1.3(a,b* *). If s 2 (mod 4), then q2 has odd order in (Z=p)x , and hence in (Z=pn)x for* * all n. So = in (Z=pn)x for all n. Thus for a p-group P and V 2 Repk(P * *), OV (g) = OV (gq) = OV (g-1) for all g 2 P if and only if OV (g) = OV (gq2) = OV* * (g-1) for all g. So (c) follows from Proposition 1.3(b). Upon combining Propositions 5.2 and 5.3 with Theorem A, we see that for p odd and q prime to p, each of the fusion systems Fp(Sp2n(q)) ' Fp(SO2n+1(q)) and Fp(GO2n(q)) is isotypically equivalent to the p-fusion system of some general l* *inear group. 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Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Be* *l- laterra, Spain E-mail address: broto@mat.uab.es Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn, Denmark E-mail address: moller@math.ku.dk LAGA, Institut Galil'ee, Av. J-B Cl'ement, F-93430 Villetaneuse, France E-mail address: bobol@math.univ-paris13.fr